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Second lecture
II. Fusion and quasifission with the dinuclear
system model
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Contents

1. Introduction
2. Models for fusion with adiabatic and diabatic
   potentials
3. Dynamics of fusion in the dinuclear system model
4. Quasifission and incomplete fusion
5. Summary and conclusions

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Work of G. G. Adamian, N. V. Antonenko,
R. V. Jolos, S. P. Ivanova, A. Zubov Joint
Institute for Nuclear Research,
Dubna Collaboration with Junqing Li, Wei Zuo,
Institute of Modern Physics,
Lanzhou Enguang Zhao, Ning Wang, Shangui Zhou
Institute of Theoretical Physics, Beijing
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• Introduction
• Superheavy elements are presently produced in
  heavy ion reactions up to element 118.
• Experiments done at GSI (Darmstadt), JINR
  (Dubna), GANIL (France), LBNL (Berkeley), RIKEN
  (Japan), IMP(Lanzhou)
• Two types of reactions are successful
• Cold and hot complete fusion reactions with
  excitation energies of the compound nucleus of
  ECN of 12-15 and 32-36 MeV, respectively.

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a) Lead-based cold fusion reactions
70Zn208Pb?278112?277112n, ?1 pb ECN ? 12
MeV for Z?108 b) Hot
fusion reactions with 48Ca projectiles
48Ca244Pu?292114?2891143n, ?1 pb ECN ?
32 MeV

• Small cross sections
• Competition between complete fusion and
  quasifission

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Idea of Volkov (Dubna) to describe fusion
reactions with the dinuclear system
concept Fusion is assumed as a transfer of
nucleons (or clusters) from the lighter nucleus
to the heavier one in a dinuclear
configuration. This process is describable with
the mass asymmetry coordinate ?(A1-A2)/(A1 A2).
A1
A2
If A1 or A2 get small, then ??1 and the system
fuses.
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The dinuclear system model uses two main
degrees of freedom to describe the fusion
and quasifission processes

• Relative motion of nuclei, capture
  of target and projectile into dinuclear system,
  decay of the
  dinuclear system quasifission
  
  
  
• Transfer of nucleons between nuclei, change
  of mass and charge asymmetries leading to fusion
  and quasifission

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Aim of lecture

• Consideration of the fusion dynamics,
• comparison of models in the calculation of
  residue cross sections for superheavy nuclei,
• discussion of quasifission with master equations,
  
• incomplete fusion as a signature for the
  dinuclear model.

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2. Models for fusion with adiabatic and diabatic
potentials

• Heavy and superheavy nuclei can be produced
  by fusion reactions with heavy ions.
• Two main collective coordinates are used for
  the description of the fusion process
• Relative internuclear distance R
• Mass asymmetry coordinate h for transfer

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Description of fusion dynamics depends strongly
whether adiabatic or diabatic potential energy
surfaces are assumed.
V
diabatic
adiabatic
R
touching configuration
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Diabatic potentials are repulsive at smaller
internuclear distances RltRt. Explanation with
two-center shell model
?i
?i
2
2
1
1
R
R
adiabatic model diabatic model
Velocity between nuclei leads to diabatic
occupation of single-particle levels,
Pauli principle between nuclei
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diabatic
adiabatic
lR
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• Models using adiabatic potentials
• Minimization of potential energy, essentially
  adiabatic potential in the internuclear distance,
  nuclei melt together.

R 0
Large probabilities of fusion for producing
nuclei with similar projectile and target nuclei.
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h
entrance quasifission
fusion
touching configuration
R
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48Ca 246Cm (from Zagrebaev)
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b) Dinuclear system (DNS) concept Fusion by
transfer of nucleons between the nuclei (idea of
V. Volkov, also von Oertzen), mainly dynamics
in mass asymmetry degree of freedom, use of
diabatic potentials, e.g. calculated with the
diabatic two-center shell model.
h 1
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h 1
fusion
entrance quasifission
touching configuration
to
R
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58Fe244Pu 302120
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110Pd 110Pd
TCSM
double folding
?R
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2R0?
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Calculation of diabatic potential
single particle energies
occupation numbers
Diabatic occupation numbers depend on time
De-excitation of diab. levels with relaxation
time, depending on single particle width.
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diabatic
adiabatic
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3. Dynamics of fusion in the dinuclear system
model
Evaporation residue cross section for the
production of superheavy nuclei
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• Partial capture cross section scap
• Dinuclear system is formed at the initial stage
  of the reaction, kinetic energy is transferred
  into potential and excitation energy.

V(R)
Bqfbarrier for quasifission
R
touching point
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b) Probability for complete fusion PCN DNS
evolves in mass asymmetry coordinate by diffusion
processes toward fusion and in the relative
coordinate toward the decay of the dinuclear
system which is quasifission.
V(h)
Bfus
h
hi
-1
1
Bfusinner fusion barrier
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222Th
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Competition between fusion and quasifission, both
processes are treated simultaneously.
Calculation of PCN and mass and charge
distributions in h and R Fokker-Planck equation,
master equations, Kramers approximation.
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c) Kramers formula for PCN Rate for fusion ??
fus Rate for quasifission ?qf ?R ?? sym,
i.e. decay in R and diffusion in ? to more
symmetric DNS.
Cold fusion (Pb-based reactions) Hot fusion
(48Ca projectiles)
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d) Importance of minima in the driving potential
Shell and deformation effects lead to local
minima in V(?). In liquid drop model the mean
value of ? runs to symmetric fragmentation ???,
then quasifission. Minima in V(?) hinder motion
to ???, then larger probability for
fusion. Idea of A. Sandulescu and W. Greiner
(1976) for optimum selection of target and
projectile choice of fragmentations in minima of
V(?).
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A. Sandulescu and W. Greiner (1976)
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114
110
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104
112
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Z114
Z108
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Example 54Cr 208Pb ? 262Sg
V(h)
Bfus
B?sym
h
54Cr 208Pb
Bfus5.7 MeV, B?sym3.6 MeV, Bqf2.7
MeV Dependence of fusion probability PCN
on barrier height B?sym.
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Bfus
B?sym
54Cr 208Pb ? 262Sg
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e) Optimum excitation energy for Pb-based fusion
reactions
V(h)
Bfus
ECN
h
hi
-1
1
ECN Ecm Q ? V(hi) Bfus
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Cold fusion
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f) Survival probability Wsur De-excitation of
excited compound nucleus by neutron emission in
competition with fission. Other decays (g-,
a-decays) can be neglected. For Pb-based
reactions with emission of one neutron
Gn width for neutron emission
Gf fission width, Gfgtgt Gn
P1n probability of realisation of 1n
channel
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g) Results for cold fusion reactions
Cold reactions AZ208Pb superheavy n
React. Super-heavy ECN (MeV) PCN ?cap (mb) Wsur ?ERth ?ERex
50Ti 208Pb 257104 n 16.1 3x10-2 5.3 9x10-5 14.3 nb 10 nb
70Zn 208Pb 272112 n 9.8 1x10-6 3.0 6x10-4 1.8 pb 1 pb
86Kr 208Pb 293118 n 13.3 1.5x 10-10 1.7 2x10-2 5.1 fb lt 0.5pb
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nb
pb
fb
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110
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112
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68
70
114
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73
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76Ni208Pb
64Ni208Pb
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ANi208Pb?110
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h) Hot fusion reactions with 48Ca projectiles and
actinide nuclei as target
e.g. 48Ca244Pu?288,289114 4,3 n,
x ?3n , ?4n 1 pb Initial dinuclear system
is more asymmetric than in Pb-based reactions
Bfus is smaller. Larger fusion probability, but
smaller survival probability because of higher
excitation energy ECN.
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Reaction Super-heavy ECN (MeV) ?xnth (pb) ?xnexp (pb)
48Ca 238U 283112 3n 33 1.7 1 - 5
48Ca 244Pu 288114 4n 35.5 1.0 1
48Ca 242Pu 286114 4n 32 1.0 2.5
48Ca 248Cm 292116 4n 35 0.2 0.5
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capture
fusion
PCN
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Z112
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Z114
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Z116
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Z118
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4. Quasifission and incomplete iiiiifusion a)
Master equation for mass and charge
transfer Probability PZN(t) to find the dinuclear
system in fragmentation Z1Z, N1N, Z2Ztot-Z1,
N2Ntot-N1 Z1N1A, Z1Z2N1N2Atot
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Starting point shell model Hamiltonian
in second quantization
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Rates D depend on single-particle energies and
temperature related to excitation energy. Only
one-nucleon transitions are assumed. LqfZ,N
rate for quasifission LfisZ,N rate for
fission of heavy nucleus
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Transfer rates
for example
?t 10-22 s transition time of
nucleon, temperature ?T(Z,N), nA1?(T), nA2?(T)
Fermi occupation numbers ?A1?, ?A2? single
particle energies.
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b) Fusion probability and mass and charge yields
Fusion probability
t0 reaction time (3-5)x10-20 s
yield of quasifission
Reaction time determined by
mass yield charge yield
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c) Total kinetic energy
Total kinetic energy distribution and its
dispersion depend on deformation of
fragments. Strong polarisation effects for
symmetric DNS for AAtot/2 ?20 3-4 times larger
deformations than in ground state.
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Distribution of clusters in charge, mass and
deformation in calculation of TKE
with frequency ?vib and stiffness parameter Cvib
of quadrupole vibrations.
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d) Hot fusion reactions with 48Ca
beam Quasifission measurements by Itkis et
al. e.g. 48 Ca 248Cm
296116 Quasifission events near initial mass
overlap with deep inelastic collision events and
are taken out in the experimental analysis. In
our calculations only quasifission is shown.
Calculated primary fragment distributions. Maxima
in yields arise from minima in driving potential
U(h).
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ECN 33.4 MeV
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286112
70
ECN42 MeV
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J70
J0
Sn
Pb
Zr
ECN 37 MeV
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Minima in variance s2TKE are related to stiff
nuclei Zr, Sn, Pb. Importance of fluctuations
of deformations for variance s2TKE .
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deformation
nucleon exchange
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58Fe248Cm?306122
ECN 33 MeV
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e) Competition between fusion-fission and
quasifission processes
Ratio of fusion-fission to quasifission events
ECN(MeV) r
40Ar 165Ho 89 120 1.1 0.7
48Ca 244Pu 34.8 50 1.4 x 10-2 1.1 x 10-1
86Kr 208Pb 17 30 2.1 x 10-7 2.0 x 10-5
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f) Incomplete fusion Transfer cross sections to
more asymmetric systems Cross section of the
production of primary heavy nucleus
Cross section with evaporation of x neutrons
Here 48Ca 244,246,248Cm, 76Ge 208Pb
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48Ca244Cm 48Ca246Cm 48Ca248Cm
Charge number of heavy nucleus
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48Ca244Cm ? Ecm207 MeV 48Ca246Cm ? Ecm205.5
MeV 48Ca248Cm ? Ecm204 MeV
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nb
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66,68,70Zn 208Pb
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• 5. Summary and conclusions
• The concept of the nuclear molecule or dinuclear
  system describes nuclear structure phenomena
  connected with cluster structures, the fusion of
  heavy nuclei to superheavy nuclei, the competing
  quasifission and fission.
• The dynamics of the dinuclear system has two main
  degrees of freedom the relative
  motion of the nuclei and the mass asymmetry
  degree of freedom.

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• Superheavy nuclei are produced in heavy ion
  collisons in a process of three stages
• (1) Capture of nuclei in a dinuclear system,
• (2) Fusion to compound nucleus by nucleon
  transfer between touching nuclei in competition
  with quasifission, which is the decay of the
  dinuclear system,
• (3) De-excitation of compound nucleus by neutron
  emission into the ground state of superheavy
  nucleus.

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• The dinuclear system has a repulsive, diabatic
  potential energy surface towards smaller
  internuclear distances. This
  potential hinders the nuclei to amalgamate
  directly.
• The transfer of nucleons and the quasifission are
  statistical diffusion processes in the excited
  dinuclear system. They are described by
  Fokker-Planck or master equations or simply by
  the Kramers approximation. Fusion and
  quasifission are simultaneously treated.

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• A precise prediction of optimum production cross
  sections for superheavy nuclei can be obtained
  from the calculation of isotopic series of
  reactions. The cross sections depend sensitively
  on the interplay between fusion and survival
  probabilities.
• For elements Z gt 118 the production cross
  sections seem to be lower than 0.1 pb. The
  border of 0.1 pb is the limit of measurements in
  the near future.

D.G.